Free-Convective Heat Transfer: With Many Photographs of Flows and Heat Exchange

519 Pages·2005·24.603 MB·English

by  Professor Oleg G. Martynenko, Dr. Pavel P. Khramtsov (auth.)

Free-ConvectiveHeatTransfer
Oleg G. Martynenko Pavel P. Khramtsov
Free-Convective
Heat Transfer
With Many Photographs of
Flows and Heat Exchange
ABC
ProfessorOlegG.Martynenko Dr.PavelP.Khramtsov
BelarusAcademyofSciences InternationalCenterofExcellence
HeatandMassTransferInstitute forResearchEng.andTechnology(ICERET)
P.Brovkastr.15 AufdemGossberg
220072Minsk 55471Wüschheim
Belarus Germany
Email:[email protected]
BelarusAcademyofSciences
HeatandMassTransferInstitute
PhysicalandChemical
HydrodynamicsLaboratory
P.Brovkastr.15
220072Minsk
Belarus
LibraryofCongressControlNumber:2005921210
ISBN-10 3-540-25001-8SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-25001-2SpringerBerlinHeidelbergNewYork
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TheauthorsexpressgratitudetoIrinaA.ShikhandTatyanaA.Baranovafor
their help in preparing and editing the text of the book and also for advice
concerningitsstyleandpresentation.ThanksalsotoNatalyaK.Shveevaand
GretaR.MaljavskayafortheirhelpineditingtheEnglishversionofthebook.
TheauthorsacknowledgethehelpfromSergeyV.VolkovandVictorS.Burak
in carrying out the research work.
Oleg G. Martynenko
Pavel P. Khramtsov
Contents
1 Basic Statements and Equations of Free Convection ....... 1
1.1 Equations and Uniqueness Conditions...................... 2
1.2 Boussinesq Approximation................................ 4
1.3 Method of Generalized Variables .......................... 6
1.4 Dimensional Analysis .................................... 9
1.5 Free-Convective Boundary Layer .......................... 12
1.6 Integral Methods ........................................ 18
1.7 Loss of Stability and Transition to Turbulence .............. 23
1.8 Outer and Inner Flow Regions ............................ 37
1.9 Experimental Methods in Free Convection .................. 38
1.10 Processing of Experimental
and Calculated Data on Heat Transfer ..................... 67
1.11 Basic Similarity Criteria and Parameters
of Free-Convective Heat Transfer .......................... 69
References .................................................. 76
2 Free Convection on a Plane................................ 81
2.1 Vertical Flat Plate
with a Constant Wall Temperature ........................ 88
2.2 Constant Heat Flow on a Vertical Plane Surface............. 94
2.3 Plane Vertical Plate with a Variable Surface Temperature .... 97
2.4 Plane Vertical Plate with a Variable Heat Flux on a Surface ..104
2.5 Free Convection on a Vertical Surface
in Stratified Media ......................................114
2.6 Conjugated Problems on Vertical Surface...................128
2.7 Discontinuity of Boundary Conditions
on the Vertical Surface ...................................139
2.8 Free Convection Near a Vertical Surface
in a Variable Field of Mass Forces .........................163
2.9 Free-Convective Heat Transfer
on a Plane Inclined Surface ...............................166
VIII Contents
2.10 Horizontal and Almost Horizontal Surfaces .................176
2.11 Spatial Flow on a Plane Surface ...........................195
2.12 Compressibility and Variability
of Thermophysical Properties .............................198
2.13 Energy Dissipation and the Work of Compression............204
2.14 Effect of Volumetric Heat Generation
on Free Convection ......................................208
2.15 Injection and Suction on a Plane Surface ...................208
References ..................................................212
3 Free Convection on Curved Surfaces.......................219
3.1 Vertical Cylinder ........................................220
3.2 Horizontal Cylinder......................................229
3.3 Inclined Cylinder ........................................243
3.4 Cone...................................................247
3.5 Vertical Needle..........................................252
3.6 Cylinder of Arbitrary Cross Section and Prism ..............252
3.7 Sphere and Spheroid.....................................257
3.8 Curved Surface of Complex Geometry......................263
References ..................................................274
4 Natural Convection in Enclosures..........................279
4.1 Spherical and Cylindrical Cavities .........................279
4.2 Rectangular Cavities and Interlayers .......................291
4.3 Cylindrical Interlayers....................................310
4.4 Spherical Interlayers .....................................322
4.5 Cavities of Complex Geometry ............................324
References ..................................................338
5 Free Convection in Tubes and Channels,
on Ribbed Surfaces and in Tube Bundles ..................345
5.1 Rectangular Tube .......................................345
5.2 Cylindrical Channel .....................................359
5.3 Finned Surfaces .........................................364
5.4 Tube Bundles ...........................................376
5.5 Panels with Cellular Grids................................384
References ..................................................388
6 Nonstationary Processes of Free Convection ...............393
6.1 Main Dependences for Calculation
of Unsteady Free Convection..............................393
6.2 Free Convection in Oscillating Flows.......................411
References ..................................................425
Contents IX
7 Heat Transfer by Mixed Convection .......................429
7.1 Effect of Radiation on Free-Convective Heat Transfer ........429
7.2 Combined Free and Forced Convection .....................434
References ..................................................471
8 Heat Transfer in Media
with Special Properties ....................................475
8.1 Water at Extreme Density................................475
8.2 Critical and Supercritical State of a Substance ..............486
8.3 Rarefied Gases and Evacuated Liquids .....................495
8.4 Convection Induced by Radiation..........................499
8.5 Biosystems .............................................501
8.6 Solidifying Melt .........................................504
References ..................................................512
1
Basic Statements and Equations
of Free Convection
A body which is brought into a fluid having another temperature is a source
of disturbance of the equiprobable state of the medium. The elements of the
fluidborderingonthebodysurfaceassumeitstemperature,andtheprocessof
heatdistribution inthefluidbymolecularthermalconductivity –theprocess
ofconduction–begins.Forasmalldifferenceoftemperatures,thisisthebasic
mechanism of heat transfer. The arising temperature nonuniformity which is
connected with the nonuniformity of density ∆ρ leads to the occurrence of
upward(downward)flowsorconvectionwhichtransfersheatfromtheobject.
In general, natural-convection heat transfer occurs in a nonuniform field
of mass forces:
→ → → →
∆F =∆(ρg )=∆ρg +ρ∆g .
t t t
If the density nonuniformity ∆ρ is due to the temperature nonuniformity,
then the occuring motion is referred to as thermal gravitational convection.
A change in the density can also be due to nonuniform distribution of the
concentrationofanymixturecomponentortochemicalreactions(inthiscase
we speak of concentration diffusion, or convection), to the presence of phases
with differentdensities ortosurfacetension forcesatthephaseinterface, etc.
Natural-convection flows can be induced by both gravitational and other
mass forces (centrifugal, Coriolis, electromagnetic, etc.). For example, in ro-
tating gas-filled channels the nonuniform mass force field is caused not only
by density difference, but also by a nonuniform acceleration field.
Motion and heat exchange occurring in an infinite space are called free
convection.Thepressureinthefieldofthermalnonuniformityandinthezone
of convective flow can be considered constant. Motion and heat exchange in
a bounded volume is called natural convection.
Natural-convectionflowscanbelaminarandturbulent.Experimentaldata
show that in free convection the basic area of thermal and hydrodynamic dis-
turbances is concentrated in a rather thin boundary layer of fluid near the
heat transfer surface. For example, at the bottom of a heated vertical plate
a laminar boundary layer is formed. With increase in the height of the plate
2 1 Basic Statements and Equations of Free Convection
the boundary layer thickness increases and thus heat transfer decreases. At
a certain height, the laminar motion is disturbed and becomes turbulent. In
this region, the flow represents random motion of the masses of the fluid
whose characteristics are described by stochastic functions of space and time
variables. For a part of the heated surface, where the characteristics of ther-
mal turbulence become statistically identical, the heat transfer coefficient is
independent of the body dimensions.
For free convection one cannot consider separately thermal and hydrody-
namic boundary layers, since the motion of fluid is fully determined by the
process of heat transfer.
1.1 Equations and Uniqueness Conditions
To describe free-convective motion and heat transfer, the laws of momentum,
mass, and energy conservation in the fluid moving under the action of mass,
surface, and inertial forces are used. In a rectangular coordinate system the
conservation equations have the form [1.1]
(cid:1) (cid:2) (cid:3) (cid:1) (cid:2)(cid:4)
∂u ∂u ∂ ∂u ∂u
ρ i +u i =ρF + µ i + j
∂τ j∂x i ∂x ∂x ∂x
j (cid:1)j (cid:2)j i
2 ∂ ∂u ∂p
− µ j − , (1.1.1)
3∂x ∂x ∂x
j j i
∂ρ ∂
+ (ρu )=0, (1.1.2)
∂τ ∂x j
j
(cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2)
∂T ∂T ∂p ∂p ∂ ∂T
ρc +u =Q +βT +u + λ
p ∂τ j∂x ν ∂τ j∂x ∂x ∂x
j (cid:5) (cid:1) (cid:2)j (cid:1) j (cid:2) (cid:6)j
1 ∂u ∂u 2 2 ∂u 2
+ µ i + j − j . (1.1.3)
2 ∂x ∂x 3 ∂x
j i j
Closure of system (1.1.1)–(1.1.3) is achieved through the thermodynamic
equation of state
ρ=ρ(p,T) (1.1.4)
and the equations relating the coefficients of viscosity, heat capacity, heat
conduction, and volumetric expansion to pressure and temperature.
Thesystemofdifferentialequationsshouldbeaugmentedwithuniqueness
conditions to single out the considered process from the whole class of the
phenomena described by the system of differential equations (1.1.1)–(1.1.3).
The geometric conditions specify the form and the linear dimensions of
the body in which the process proceeds.
1.1 Equations and Uniqueness Conditions 3
Initial conditions are necessary in the problems of nonstationary free con-
vection. They represent the distribution of velocities and temperatures at a
certain initial moment of the time τ =τ :
0
u =u (x ),
i i0 i
(1.1.5)
T =T (x ).
0 i
Boundary conditions specify the values of the required functions at the
boundaries of the region considered and can be described in a number of
ways.
Forasolidbodyinaviscousfluidflow,whenthefreepathofmoleculesin
the fluid is mach smaller than the characteristic size of the body, the velocity
of the particles of the fluid on a fixed surface is equal to zero, whereas on a
moving one it coincides with the velocity of the points of the surface (no-slip
condition):
u (τ,x )=0. (1.1.6)
iw iw
Inweaklyrarefiedgasesthesleepvelocityonasolidsurfaceisproportional
tothederivativeofthetangentvelocitycomponentwithrespecttothenormal
to the surface. If there is mass transfer on the surface, the normal velocity
component is determined by the rate of absorption (release) of substance by
the wall. The boundary conditions also include setting specification of the
velocity far from the body immersed in a flow. A large variety of boundary
conditions exist for temperature.
The boundary condition of the 1st kind consists in specification of tem-
perature distribution over the heating surface at any instant of time:
T =T (τ,x ) . (1.1.7)
w iw
The boundary condition of the 2nd kind specifies the heat flux density for
each point of the body surface as a function of time:
(cid:1) (cid:2)
∂T
−λ =q (τ,x ) , (1.1.8)
jw ∂n jw iw
j
where j is the number of continuous boundary surfaces of the body.
The simplest boundary conditions are the constancy of temperature or of
the heat flux density on the body surface:
T =T =const,
(cid:1) (cid:2)w
∂T
−λ =q =const. (1.1.9)
jw ∂n w
j w
Theboundaryconditionofthe3rdkindcharacterizesthelawofconvective
heat exchange between the body surface and environment. In this case, the
heat flux density is directly proportional to the difference of temperatures
between the body surface and the environment: